{"id":144,"date":"2018-07-23T05:44:18","date_gmt":"2018-07-23T05:44:18","guid":{"rendered":"http:\/\/csp11.epgpbooks.inflibnet.ac.in\/?post_type=chapter&#038;p=144"},"modified":"2018-12-27T11:17:01","modified_gmt":"2018-12-27T11:17:01","slug":"exponentiation-and-logarithm","status":"publish","type":"chapter","link":"https:\/\/ebooks.inflibnet.ac.in\/csp11\/chapter\/exponentiation-and-logarithm\/","title":{"rendered":"Exponentiation and Logarithm"},"content":{"raw":"<div><span style=\"float: right\"><a href=\"https:\/\/youtu.be\/f1chUP70-Gk\" target=\"_blank\" rel=\"noopener\"><img src=\"http:\/\/epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/2018\/11\/download.png\" alt=\"epgp books\" width=\"75px\" height=\"75px;\" \/><\/a>\r\n<\/span><\/div>\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\n<strong>Learning Objectives<\/strong>\r\n<ul>\r\n \t<li>To discuss about Exponentiation and Logarithm algorithm<\/li>\r\n \t<li>To understand the need for Order of group<\/li>\r\n \t<li>To Know about the purpose of Primitive root<\/li>\r\n \t<li>To discuss various examples related to the topic<\/li>\r\n<\/ul>\r\n<strong>\u00a0 \u00a012.1 Exponentiation and Logarithm<\/strong>\r\n\r\n&nbsp;\r\n<p style=\"text-align: justify\">Exponentiation and logarithms are inverse of each other. The following shows the relationship between them, in which a is called the base of the exponentiation or logarithm.<\/p>\r\nExponentiation:\u00a0<img class=\"alignnone size-full wp-image-145\" src=\"http:\/\/csp11.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/55\/2018\/07\/Capture.jpg\" alt=\"\" width=\"342\" height=\"35\" \/>\r\n\r\n&nbsp;\r\n\r\n<strong>12.2 Exponentiation<\/strong>\r\n\r\n&nbsp;\r\n\r\nIn cryptography, a common modular operation is exponentiation that is we often need to calculate\r\n\r\n&nbsp;\r\n\r\n<img class=\"size-full wp-image-146 aligncenter\" src=\"http:\/\/csp11.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/55\/2018\/07\/Capture-1.jpg\" alt=\"\" width=\"104\" height=\"40\" \/>\r\n<div>\r\n<p style=\"text-align: justify\">The RSA cryptosystems are exponentiation for both encryption and decryption with very large exponents. Unfortunately, most computer languages have no operator that can efficiently compute exponentiation, particularly then the exponent is very large. To make this type of calculation more efficient, we need algorithms that are more efficient.<\/p>\r\n\r\n<\/div>\r\n<strong>\u00a0 12.3 Logarithm<\/strong>\r\n\r\n&nbsp;\r\n<p style=\"text-align: justify\">In cryptography, we also need to discuss modular logarithm. If we use exponentiation to encrypt or decrypt, the adversary can use logarithm to attack. We need to know how hard it is to reverse the exponentiation.<\/p>\r\n&nbsp;\r\n\r\n<strong>Executive search<\/strong>\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\nThe first solution that might come to mind is to solve x=log a y (mod n). We can write an algorithm\r\n\r\n&nbsp;\r\n\r\nthat continuously calculates y=ax mod n until it finds the value of given y. Algorithm shown below\r\n\r\n&nbsp;\r\n\r\nshows this approach.\r\n\r\n&nbsp;\r\n\r\nModular _Logarithm (a ,y, n)\r\n\r\n&nbsp;\r\n\r\n{\r\n\r\n&nbsp;\r\n\r\nFor (x=1 to n-1)\u00a0 \/\/ k is the number of bits in x\r\n\r\n&nbsp;\r\n\r\n{\r\n\r\n&nbsp;\r\n\r\nIf (y =\u00a0\u00a0\u00a0 mod n) return x\r\n\r\n&nbsp;\r\n\r\n}\r\n\r\n&nbsp;\r\n\r\nReturn failure\r\n\r\n&nbsp;\r\n\r\n}\r\n\r\n&nbsp;\r\n<div>\r\n\r\n<strong>12.3.1 Discrete Logarithm<\/strong>\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n<p style=\"text-align: justify\">The second approach is to use the concept of discrete logarithm. Understand this concept requires understanding some properties of multiplicative groups. In mathematics, a discrete logarithm is an integer k exponent solving the equation bk = g, where b and g are elements of a group. Discrete logarithms are logarithms defined with regard to multiplicative cyclic groups. If <em>G<\/em> is a multiplicative cyclic group and <em>g<\/em> is a generator of <em>G<\/em>, then from the definition of cyclic groups, we know every element <em>h<\/em> in <em>G<\/em> can be written as <em>g<\/em><sup><em>x<\/em><\/sup> for some <em>x<\/em>. The discrete logarithm to the base <em>g<\/em> of <em>h<\/em> in the group <em>G<\/em> is defined to be <em>x<\/em>.<\/p>\r\n&nbsp;\r\n<p style=\"text-align: justify\">For example, if the group is <em>Z<\/em>5*, and the generator is 2, then the discrete logarithm of 1 is 4 because 2<sup>4<\/sup> \u2261 1 mod 5.<\/p>\r\n&nbsp;\r\n<p style=\"text-align: justify\">Fix a prime p. Let a, b be nonzero integers (mod p). The problem of finding x such that ax \u2261 b (mod p) is called the discrete logarithm problem. Suppose that n is the smallest integer such that an \u22611<\/p>\r\n(mod p), i.e., n=ord(a). By assuming 0\u2264x&lt;n, we denote x=La(b), and call it the discrete log of b w.r.t. a (mod p)\r\n\r\n&nbsp;\r\n\r\nExample: p=11, a=2, b=9, then x=L2(9)=6\r\n\r\n<\/div>\r\n<strong>Finite Multiplicative Group<\/strong>\r\n\r\n&nbsp;\r\n<p style=\"text-align: justify\">In cryptography, we often use the multiplicative finite group: G=&lt;Zn*,x&gt;in which the operation is multiplication. The set Zn*contains those integers from 1 to n-1 that are relatively prime to n; the identity element is e=1. Note that when the modulus of the group is a prime, we have G=&lt;Zp*,x&gt;. This group is the special case of the first group, so we concentrate on the first group in this section.<\/p>\r\n&nbsp;\r\n\r\n<strong>12.3.2 Cyclic groups and generators<\/strong>\r\n\r\n&nbsp;\r\n<p style=\"text-align: justify\">Some groups have an interesting property: all the elements in the group can be obtained by repeatedly applying the group operation to a particular group element. If a group has such a property, it is called a cyclic group and the particular group element is called a generator. A trivial example is the group Zn, the additive group of integers modulo n. In Zn, 1 is always a generator:<\/p>\r\n&nbsp;\r\n<p style=\"text-align: center\">1 \u2261 1 mod n<\/p>\r\n<p style=\"text-align: center\">1+1 \u2261 2 mod n<\/p>\r\n<p style=\"text-align: center\">1+1+1 \u2261 3 mod n<\/p>\r\n<p style=\"text-align: center\">...<\/p>\r\n<p style=\"text-align: center\">1+1+1+...+1 \u2261 n \u2261 0 mod n<\/p>\r\n&nbsp;\r\n<p style=\"text-align: justify\">If a group is cyclic, then there may exist multiple generators. For example, we know <em>Z<\/em>5 is a cyclic group. The element 1 is a generator for sure. And if we take a look at 2, we can find:<\/p>\r\n\r\n<div>\r\n\r\n&nbsp;\r\n<p style=\"text-align: center\">2 \u2261 2 mod 5<\/p>\r\n<p style=\"text-align: center\">2+2 \u2261 4 mod 5<\/p>\r\n<p style=\"text-align: center\">2+2+2 \u2261 6 \u2261 1 mod 5<\/p>\r\n<p style=\"text-align: center\">2+2+2+2 \u2261 8 \u2261 3 mod 5<\/p>\r\n<p style=\"text-align: center\">2+2+2+2+2 \u2261 10 \u2261 0 mod 5<\/p>\r\n\r\n<\/div>\r\n&nbsp;\r\n<p style=\"text-align: justify\">So all the group elements {0, 1, 2, 3, 4} in <em>Z<\/em>5 can also be generated by 2. That is to say, 2 is also a generator for the group <em>Z<\/em>5<\/p>\r\n&nbsp;\r\n\r\n<strong>Order of the Group<\/strong>\r\n\r\n&nbsp;\r\n<p style=\"text-align: justify\">The order of the finite group. |G|, to be the number of elements in the group G. In G=&lt;Zn*,x&gt;, it can be proven that the order of group is \u0278(n). We have shown how to calculate \u0278(n), when n can be forced into primes.<\/p>\r\nThere are 12 elements in this group: 1, 2, 4, 5, 8, 10, 11, 13, 16, 17, 19, and 20. All are relatively prime with 21.\r\n\r\n&nbsp;\r\n\r\n<strong>Order of Element<\/strong>\r\n\r\n&nbsp;\r\n<p style=\"text-align: justify\">In G=&lt;Zn*,x&gt;, we continue with the same definition. The order of an element, a , is the smallest integer i such that a<sup>i<\/sup>\u2261e(mod n). The identity element e is 1 in this case<\/p>\r\nExample : Find the order of all elements in G = &lt;Z10\u2217, \u00d7&gt;.\r\n\r\n&nbsp;\r\n<p style=\"text-align: justify\">This group has only f(10) = 4 elements: 1, 3, 7, 9. We can find the order of each element by trial and error.<\/p>\r\n\r\n<div>\r\n\r\n\u00a0 \u00a0 a. 1<sup>1<\/sup> \u2261 1 mod (10) \u2192 ord(1) = 1.\r\n\r\n&nbsp;\r\n\r\nb. 3<sup>4<\/sup> \u2261 1 mod (10) \u2192 ord(3) = 4.\r\n\r\n&nbsp;\r\n\r\nc. 7<sup>4<\/sup> \u2261 1 mod (10) \u2192 ord(7) = 4.\r\n\r\n&nbsp;\r\n\r\nd. 9<sup>2<\/sup> \u2261 1 mod (10) \u2192 ord(9) = 2.\r\n\r\n<\/div>\r\n<strong>12.3.6 Primitive roots<\/strong>\r\n\r\n&nbsp;\r\n<p style=\"text-align: justify\">A very interesting concept in multiplicative group is that of primitive root, which is used in the ElGamal cryptosystems. In the group G = &lt;Zn\u2217, \u00d7&gt;, when the order of an element is the same as \u0278(n), that element is called the primitive root of the group.<\/p>\r\n&nbsp;\r\n<p style=\"text-align: justify\">Example: There are no primitive roots in G = &lt;Z8\u2217, \u00d7&gt; because no element has the order equal to f (8) = 4. The order of elements are all smaller than 4.<\/p>\r\n<p style=\"text-align: justify\">Example: shows the result of <em>a<\/em><sup><em>i<\/em> <\/sup>\u2261 <em>x<\/em> (mod 7) for the group G = &lt;Z7\u2217, \u00d7&gt;. In this group, f(7) = 6.<\/p>\r\n&nbsp;\r\n\r\n<img class=\"alignnone size-full wp-image-147\" src=\"http:\/\/csp11.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/55\/2018\/07\/Capture-2.jpg\" alt=\"\" width=\"747\" height=\"268\" \/>\r\n<p style=\"text-align: justify\">The orders of elements are ord(1)=1, ord(2)=3, ord(3)=6, ord(5)=6, and ord(6)=1.The table 37.1 shows that only two elements, 3 and 5, have the order at i = \u0278(n)=6.Therefore, this group has only two primitive roots: 3 and 5.<\/p>\r\n&nbsp;\r\n<p style=\"text-align: justify\">It has been proved that the group G = &lt;Z<em>n<\/em>*, \u00d7&gt; has primitive roots only if <em>n<\/em> is 2, 4, <em>p<\/em>t, or 2<em>p<\/em>t.in which p is an odd prime (not 2) and t is an integer.<\/p>\r\n\r\n<div>\r\n\r\nExample: For which value of <em>n<\/em>, does the group G = &lt;Z<em>n<\/em>\u2217, \u00d7&gt; have primitive roots: 17, 20, 38, and 50?\r\n\r\na.\u00a0 \u00a0 \u00a0 \u00a0 G = &lt;Z17\u2217, \u00d7&gt; has primitive roots, 17 is a prime.\r\n\r\nb.\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 G = &lt;Z20\u2217, \u00d7&gt; has no primitive roots.\r\n\r\nc.\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 G = &lt;Z38\u2217, \u00d7&gt; has primitive roots, 38 = 2 \u00d7 19 prime.\r\n\r\n<span style=\"font-size: 1em\">d.\u00a0 \u00a0 \u00a0 \u00a0 G = &lt;Z50\u2217, \u00d7&gt; has primitive roots, 50 = 2 \u00d7 5<sup>2<\/sup> and 5 is a prime.<\/span>\r\n\r\n<\/div>\r\n<p style=\"text-align: justify\">If a group has a primitive root, then it normally has several of them. The number of primitive roots can be calculated as \u0278(\u0278(n)). For example, the number of primitive roots of<\/p>\r\n&nbsp;\r\n<p style=\"text-align: justify\">G=&lt;Z17*,x&gt;is\u0278(\u0278(17))=\u0278(16)=8.Note that we should first check to see if the group has any primitive root, before we find the number of roots.<\/p>\r\n&nbsp;\r\n<p style=\"text-align: justify\">If the group G=&lt;Z<em>n<\/em>*, \u00d7&gt; has primitive roots, the number of primitive roots is \u0278(\u0278(n)).<\/p>\r\n&nbsp;\r\n\r\n<strong>12.3.7 Cyclic Group<\/strong>\r\n\r\n&nbsp;\r\n<p style=\"text-align: justify\">If the group G=&lt;Z<em>n<\/em>*, \u00d7&gt; has primitive roots, it is cyclic. Each primitive root is a generator and can be used to create the whole set. In other words, if g is a primitive root in the group, we can generate the set Zn*as<\/p>\r\n<p style=\"text-align: justify\">as Zn\u2217 = {g1, g2, g3, \u2026, gf(n)}<\/p>\r\n&nbsp;\r\n\r\nExample:\r\n<p style=\"text-align: justify\">The group G = &lt;Z10*, \u00d7&gt; has two primitive roots because f(10) = 4 and f(f(10)) = 2. It can be found that the primitive roots are 3 and 7. The following shows how we can create the whole set Z10* using each primitive root.<\/p>\r\n<img class=\"alignnone size-full wp-image-148 aligncenter\" src=\"http:\/\/csp11.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/55\/2018\/07\/Capture-3.jpg\" alt=\"\" width=\"614\" height=\"57\" \/>\r\n<p style=\"text-align: justify\">Note that the group G = &lt;Z10*, \u00d7&gt; is always cyclic because p is a prime. The group G =&lt; Z n *, X &gt; is a cyclic group if it has primitive roots. The group G =&lt; Z p*, X &gt; is always cyclic .<\/p>\r\n&nbsp;\r\n<p style=\"text-align: justify\"><strong>The idea of Discrete Logarithm<\/strong><\/p>\r\n&nbsp;\r\n\r\nProperties of G = &lt;Zp*, \u00d7&gt; :\r\n<div>\r\n\r\n\u00a0 \u00a0 1.\u00a0 Its elements include all integers from 1 to p \u2212 1.\r\n\r\n&nbsp;\r\n\r\n2.\u00a0 It always has primitive roots.\r\n\r\n&nbsp;\r\n\r\n3.\u00a0 It is cyclic. The elements can be created using gx where x is an integer from 1 to f(n) = p \u2212 1.\r\n\r\n<\/div>\r\n<ol start=\"4\">\r\n \t<li style=\"text-align: justify\">The primitive roots can be thought as the base of logarithm. If the group has k primitive roots, calculation can be done in k different base. Given x=log, y for any element y is the set, there is another element x that is the log of y in base g. this type of logarithm is called discrete logarithm.<\/li>\r\n<\/ol>\r\n<strong>\u00a0 \u00a0Solution to Modular Logarithm Using Discrete Logs<\/strong>\r\n\r\n&nbsp;\r\n\r\nNow let us see how to solve problems of type y=ax (mod n) when y is given and we need to find x\r\n\r\n&nbsp;\r\n\r\n<strong>Tabulation of Discrete Logarithms<\/strong>\r\n\r\n&nbsp;\r\n<p style=\"text-align: justify\">One way to solve the above mentioned problem is to use a table for each Zp* and different bases.This type of table can be pre calculated and saved. For example, the below table shows the tabulation of the discrete logarithm for Z7*. We know that we have two primitive roots or base in the set.<\/p>\r\n&nbsp;\r\n\r\n<img class=\"alignnone size-full wp-image-150\" src=\"http:\/\/csp11.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/55\/2018\/07\/Capture-4.jpg\" alt=\"\" width=\"689\" height=\"97\" \/>\r\n\r\n&nbsp;\r\n\r\n<strong>Table 12.1 Discrete logarithm for G = &lt;Z<\/strong><strong>7<\/strong><strong>*, \u00d7&gt;<\/strong>\r\n\r\n&nbsp;\r\n\r\nExample: Find x in each of the following cases:\r\n<ol>\r\n \t<li>4 \u2261 3<sup><em>x<\/em><\/sup> (mod 7).<\/li>\r\n \t<li>6 \u2261 5<sup><em>x<\/em><\/sup> (mod 7).<\/li>\r\n<\/ol>\r\n<div>\r\n\r\nWe can easily use the tabulation of the discrete logarithm.\r\n\r\n&nbsp;\r\n\r\na. 4 \u2261 3<sup>x<\/sup> mod 7 \u2192 x = L34 mod 7 = 4 mod 7\r\n\r\n&nbsp;\r\n\r\nb. 6 \u2261 5<sup>x<\/sup> mod 7 \u2192 x = L56 mod 7 = 3 mod 7\r\n\r\n<\/div>\r\n<strong>One-way function<\/strong>\r\n<p style=\"text-align: justify\">A function f(x) is called a one-way function if f(x) is easy to compute, but, given y, it is computationally infeasible to find x with y=f(x). La(b) is a one-way function if p is large<\/p>\r\n&nbsp;\r\n\r\n<strong>Using Properties of Discrete Logarithms<\/strong>\r\n<p style=\"text-align: justify\">To see that discrete logarithms behave just like traditional logarithms, several properties of both types of logarithms are given in table 12 .3.Note that the modulus is \u03a6(n) instead of n.<\/p>\r\n&nbsp;\r\n\r\n<img class=\"size-full wp-image-151 aligncenter\" src=\"http:\/\/csp11.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/55\/2018\/07\/Capture-5.jpg\" alt=\"\" width=\"571\" height=\"295\" \/>\r\n<p style=\"text-align: center\"><strong>Table 12.3 Comparison of traditional and discrete logarithms<\/strong><\/p>\r\n&nbsp;\r\n\r\n<strong>Using Algorithms Based on Discrete Logarithms<\/strong>\r\n\r\n&nbsp;\r\n<p style=\"text-align: justify\">Tabulation and the properties of discrete logarithms cannot be used to solve y \u2261 ax (mod n) when n is very large. Several algorithms have been devised that use the basic idea of discrete logarithms to solve the problem. Although all of these algorithms are more efficient that the exhaustive-search algorithm that we mentioned at the beginning of this section, none of them have polynomial complexity. Most of these algorithms have the same level of complexity as the factorization problem.<\/p>\r\n&nbsp;\r\n<p style=\"text-align: justify\">The discrete logarithm problem has the same complexity as the factorization problem.<\/p>\r\n&nbsp;\r\n\r\n<strong>Summary<\/strong>\r\n<ul>\r\n \t<li>We have learnt about Exponentiation and\u00a0 Logarithm algorithm<\/li>\r\n \t<li>\u00a0Observed the need for Order of group and Cyclic Group<\/li>\r\n \t<li>\u00a0Learnt about the purpose of Primitive root<\/li>\r\n \t<li>\u00a0Understand the Discrete Logarithm problem<\/li>\r\n<\/ul>\r\n<table>\r\n<tbody>\r\n<tr>\r\n<td><strong>you can view video on Exponentiation and Logarithm<\/strong><\/td>\r\n<td><a href=\"https:\/\/youtu.be\/f1chUP70-Gk\" target=\"_blank\" rel=\"noopener\"><img class=\"alignnone wp-image-120\" src=\"http:\/\/epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/2018\/11\/download.png\" alt=\"\" width=\"36\" height=\"36\" \/><\/a><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n\r\n<img class=\"size-full wp-image-153 aligncenter\" src=\"http:\/\/csp11.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/55\/2018\/07\/Capture-6.jpg\" alt=\"\" width=\"467\" height=\"319\" \/>","rendered":"<div><span style=\"float: right\"><a href=\"https:\/\/youtu.be\/f1chUP70-Gk\" target=\"_blank\" rel=\"noopener\"><img decoding=\"async\" src=\"http:\/\/epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/2018\/11\/download.png\" alt=\"epgp books\" width=\"75px\" height=\"75px;\" \/><\/a><br \/>\n<\/span><\/div>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p><strong>Learning Objectives<\/strong><\/p>\n<ul>\n<li>To discuss about Exponentiation and Logarithm algorithm<\/li>\n<li>To understand the need for Order of group<\/li>\n<li>To Know about the purpose of Primitive root<\/li>\n<li>To discuss various examples related to the topic<\/li>\n<\/ul>\n<p><strong>\u00a0 \u00a012.1 Exponentiation and Logarithm<\/strong><\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\">Exponentiation and logarithms are inverse of each other. The following shows the relationship between them, in which a is called the base of the exponentiation or logarithm.<\/p>\n<p>Exponentiation:\u00a0<img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-145\" src=\"http:\/\/csp11.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/55\/2018\/07\/Capture.jpg\" alt=\"\" width=\"342\" height=\"35\" srcset=\"https:\/\/ebooks.inflibnet.ac.in\/csp11\/wp-content\/uploads\/sites\/55\/2018\/07\/Capture.jpg 342w, https:\/\/ebooks.inflibnet.ac.in\/csp11\/wp-content\/uploads\/sites\/55\/2018\/07\/Capture-300x31.jpg 300w, https:\/\/ebooks.inflibnet.ac.in\/csp11\/wp-content\/uploads\/sites\/55\/2018\/07\/Capture-65x7.jpg 65w, https:\/\/ebooks.inflibnet.ac.in\/csp11\/wp-content\/uploads\/sites\/55\/2018\/07\/Capture-225x23.jpg 225w\" sizes=\"auto, (max-width: 342px) 100vw, 342px\" \/><\/p>\n<p>&nbsp;<\/p>\n<p><strong>12.2 Exponentiation<\/strong><\/p>\n<p>&nbsp;<\/p>\n<p>In cryptography, a common modular operation is exponentiation that is we often need to calculate<\/p>\n<p>&nbsp;<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"size-full wp-image-146 aligncenter\" src=\"http:\/\/csp11.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/55\/2018\/07\/Capture-1.jpg\" alt=\"\" width=\"104\" height=\"40\" srcset=\"https:\/\/ebooks.inflibnet.ac.in\/csp11\/wp-content\/uploads\/sites\/55\/2018\/07\/Capture-1.jpg 104w, https:\/\/ebooks.inflibnet.ac.in\/csp11\/wp-content\/uploads\/sites\/55\/2018\/07\/Capture-1-65x25.jpg 65w\" sizes=\"auto, (max-width: 104px) 100vw, 104px\" \/><\/p>\n<div>\n<p style=\"text-align: justify\">The RSA cryptosystems are exponentiation for both encryption and decryption with very large exponents. Unfortunately, most computer languages have no operator that can efficiently compute exponentiation, particularly then the exponent is very large. To make this type of calculation more efficient, we need algorithms that are more efficient.<\/p>\n<\/div>\n<p><strong>\u00a0 12.3 Logarithm<\/strong><\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\">In cryptography, we also need to discuss modular logarithm. If we use exponentiation to encrypt or decrypt, the adversary can use logarithm to attack. We need to know how hard it is to reverse the exponentiation.<\/p>\n<p>&nbsp;<\/p>\n<p><strong>Executive search<\/strong><\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>The first solution that might come to mind is to solve x=log a y (mod n). We can write an algorithm<\/p>\n<p>&nbsp;<\/p>\n<p>that continuously calculates y=ax mod n until it finds the value of given y. Algorithm shown below<\/p>\n<p>&nbsp;<\/p>\n<p>shows this approach.<\/p>\n<p>&nbsp;<\/p>\n<p>Modular _Logarithm (a ,y, n)<\/p>\n<p>&nbsp;<\/p>\n<p>{<\/p>\n<p>&nbsp;<\/p>\n<p>For (x=1 to n-1)\u00a0 \/\/ k is the number of bits in x<\/p>\n<p>&nbsp;<\/p>\n<p>{<\/p>\n<p>&nbsp;<\/p>\n<p>If (y =\u00a0\u00a0\u00a0 mod n) return x<\/p>\n<p>&nbsp;<\/p>\n<p>}<\/p>\n<p>&nbsp;<\/p>\n<p>Return failure<\/p>\n<p>&nbsp;<\/p>\n<p>}<\/p>\n<p>&nbsp;<\/p>\n<div>\n<p><strong>12.3.1 Discrete Logarithm<\/strong><\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\">The second approach is to use the concept of discrete logarithm. Understand this concept requires understanding some properties of multiplicative groups. In mathematics, a discrete logarithm is an integer k exponent solving the equation bk = g, where b and g are elements of a group. Discrete logarithms are logarithms defined with regard to multiplicative cyclic groups. If <em>G<\/em> is a multiplicative cyclic group and <em>g<\/em> is a generator of <em>G<\/em>, then from the definition of cyclic groups, we know every element <em>h<\/em> in <em>G<\/em> can be written as <em>g<\/em><sup><em>x<\/em><\/sup> for some <em>x<\/em>. The discrete logarithm to the base <em>g<\/em> of <em>h<\/em> in the group <em>G<\/em> is defined to be <em>x<\/em>.<\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\">For example, if the group is <em>Z<\/em>5*, and the generator is 2, then the discrete logarithm of 1 is 4 because 2<sup>4<\/sup> \u2261 1 mod 5.<\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\">Fix a prime p. Let a, b be nonzero integers (mod p). The problem of finding x such that ax \u2261 b (mod p) is called the discrete logarithm problem. Suppose that n is the smallest integer such that an \u22611<\/p>\n<p>(mod p), i.e., n=ord(a). By assuming 0\u2264x&lt;n, we denote x=La(b), and call it the discrete log of b w.r.t. a (mod p)<\/p>\n<p>&nbsp;<\/p>\n<p>Example: p=11, a=2, b=9, then x=L2(9)=6<\/p>\n<\/div>\n<p><strong>Finite Multiplicative Group<\/strong><\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\">In cryptography, we often use the multiplicative finite group: G=&lt;Zn*,x&gt;in which the operation is multiplication. The set Zn*contains those integers from 1 to n-1 that are relatively prime to n; the identity element is e=1. Note that when the modulus of the group is a prime, we have G=&lt;Zp*,x&gt;. This group is the special case of the first group, so we concentrate on the first group in this section.<\/p>\n<p>&nbsp;<\/p>\n<p><strong>12.3.2 Cyclic groups and generators<\/strong><\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\">Some groups have an interesting property: all the elements in the group can be obtained by repeatedly applying the group operation to a particular group element. If a group has such a property, it is called a cyclic group and the particular group element is called a generator. A trivial example is the group Zn, the additive group of integers modulo n. In Zn, 1 is always a generator:<\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: center\">1 \u2261 1 mod n<\/p>\n<p style=\"text-align: center\">1+1 \u2261 2 mod n<\/p>\n<p style=\"text-align: center\">1+1+1 \u2261 3 mod n<\/p>\n<p style=\"text-align: center\">&#8230;<\/p>\n<p style=\"text-align: center\">1+1+1+&#8230;+1 \u2261 n \u2261 0 mod n<\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\">If a group is cyclic, then there may exist multiple generators. For example, we know <em>Z<\/em>5 is a cyclic group. The element 1 is a generator for sure. And if we take a look at 2, we can find:<\/p>\n<div>\n<p>&nbsp;<\/p>\n<p style=\"text-align: center\">2 \u2261 2 mod 5<\/p>\n<p style=\"text-align: center\">2+2 \u2261 4 mod 5<\/p>\n<p style=\"text-align: center\">2+2+2 \u2261 6 \u2261 1 mod 5<\/p>\n<p style=\"text-align: center\">2+2+2+2 \u2261 8 \u2261 3 mod 5<\/p>\n<p style=\"text-align: center\">2+2+2+2+2 \u2261 10 \u2261 0 mod 5<\/p>\n<\/div>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\">So all the group elements {0, 1, 2, 3, 4} in <em>Z<\/em>5 can also be generated by 2. That is to say, 2 is also a generator for the group <em>Z<\/em>5<\/p>\n<p>&nbsp;<\/p>\n<p><strong>Order of the Group<\/strong><\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\">The order of the finite group. |G|, to be the number of elements in the group G. In G=&lt;Zn*,x&gt;, it can be proven that the order of group is \u0278(n). We have shown how to calculate \u0278(n), when n can be forced into primes.<\/p>\n<p>There are 12 elements in this group: 1, 2, 4, 5, 8, 10, 11, 13, 16, 17, 19, and 20. All are relatively prime with 21.<\/p>\n<p>&nbsp;<\/p>\n<p><strong>Order of Element<\/strong><\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\">In G=&lt;Zn*,x&gt;, we continue with the same definition. The order of an element, a , is the smallest integer i such that a<sup>i<\/sup>\u2261e(mod n). The identity element e is 1 in this case<\/p>\n<p>Example : Find the order of all elements in G = &lt;Z10\u2217, \u00d7&gt;.<\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\">This group has only f(10) = 4 elements: 1, 3, 7, 9. We can find the order of each element by trial and error.<\/p>\n<div>\n<p>\u00a0 \u00a0 a. 1<sup>1<\/sup> \u2261 1 mod (10) \u2192 ord(1) = 1.<\/p>\n<p>&nbsp;<\/p>\n<p>b. 3<sup>4<\/sup> \u2261 1 mod (10) \u2192 ord(3) = 4.<\/p>\n<p>&nbsp;<\/p>\n<p>c. 7<sup>4<\/sup> \u2261 1 mod (10) \u2192 ord(7) = 4.<\/p>\n<p>&nbsp;<\/p>\n<p>d. 9<sup>2<\/sup> \u2261 1 mod (10) \u2192 ord(9) = 2.<\/p>\n<\/div>\n<p><strong>12.3.6 Primitive roots<\/strong><\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\">A very interesting concept in multiplicative group is that of primitive root, which is used in the ElGamal cryptosystems. In the group G = &lt;Zn\u2217, \u00d7&gt;, when the order of an element is the same as \u0278(n), that element is called the primitive root of the group.<\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\">Example: There are no primitive roots in G = &lt;Z8\u2217, \u00d7&gt; because no element has the order equal to f (8) = 4. The order of elements are all smaller than 4.<\/p>\n<p style=\"text-align: justify\">Example: shows the result of <em>a<\/em><sup><em>i<\/em> <\/sup>\u2261 <em>x<\/em> (mod 7) for the group G = &lt;Z7\u2217, \u00d7&gt;. In this group, f(7) = 6.<\/p>\n<p>&nbsp;<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-147\" src=\"http:\/\/csp11.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/55\/2018\/07\/Capture-2.jpg\" alt=\"\" width=\"747\" height=\"268\" srcset=\"https:\/\/ebooks.inflibnet.ac.in\/csp11\/wp-content\/uploads\/sites\/55\/2018\/07\/Capture-2.jpg 747w, https:\/\/ebooks.inflibnet.ac.in\/csp11\/wp-content\/uploads\/sites\/55\/2018\/07\/Capture-2-300x108.jpg 300w, https:\/\/ebooks.inflibnet.ac.in\/csp11\/wp-content\/uploads\/sites\/55\/2018\/07\/Capture-2-65x23.jpg 65w, https:\/\/ebooks.inflibnet.ac.in\/csp11\/wp-content\/uploads\/sites\/55\/2018\/07\/Capture-2-225x81.jpg 225w, https:\/\/ebooks.inflibnet.ac.in\/csp11\/wp-content\/uploads\/sites\/55\/2018\/07\/Capture-2-350x126.jpg 350w\" sizes=\"auto, (max-width: 747px) 100vw, 747px\" \/><\/p>\n<p style=\"text-align: justify\">The orders of elements are ord(1)=1, ord(2)=3, ord(3)=6, ord(5)=6, and ord(6)=1.The table 37.1 shows that only two elements, 3 and 5, have the order at i = \u0278(n)=6.Therefore, this group has only two primitive roots: 3 and 5.<\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\">It has been proved that the group G = &lt;Z<em>n<\/em>*, \u00d7&gt; has primitive roots only if <em>n<\/em> is 2, 4, <em>p<\/em>t, or 2<em>p<\/em>t.in which p is an odd prime (not 2) and t is an integer.<\/p>\n<div>\n<p>Example: For which value of <em>n<\/em>, does the group G = &lt;Z<em>n<\/em>\u2217, \u00d7&gt; have primitive roots: 17, 20, 38, and 50?<\/p>\n<p>a.\u00a0 \u00a0 \u00a0 \u00a0 G = &lt;Z17\u2217, \u00d7&gt; has primitive roots, 17 is a prime.<\/p>\n<p>b.\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 G = &lt;Z20\u2217, \u00d7&gt; has no primitive roots.<\/p>\n<p>c.\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 G = &lt;Z38\u2217, \u00d7&gt; has primitive roots, 38 = 2 \u00d7 19 prime.<\/p>\n<p><span style=\"font-size: 1em\">d.\u00a0 \u00a0 \u00a0 \u00a0 G = &lt;Z50\u2217, \u00d7&gt; has primitive roots, 50 = 2 \u00d7 5<sup>2<\/sup> and 5 is a prime.<\/span><\/p>\n<\/div>\n<p style=\"text-align: justify\">If a group has a primitive root, then it normally has several of them. The number of primitive roots can be calculated as \u0278(\u0278(n)). For example, the number of primitive roots of<\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\">G=&lt;Z17*,x&gt;is\u0278(\u0278(17))=\u0278(16)=8.Note that we should first check to see if the group has any primitive root, before we find the number of roots.<\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\">If the group G=&lt;Z<em>n<\/em>*, \u00d7&gt; has primitive roots, the number of primitive roots is \u0278(\u0278(n)).<\/p>\n<p>&nbsp;<\/p>\n<p><strong>12.3.7 Cyclic Group<\/strong><\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\">If the group G=&lt;Z<em>n<\/em>*, \u00d7&gt; has primitive roots, it is cyclic. Each primitive root is a generator and can be used to create the whole set. In other words, if g is a primitive root in the group, we can generate the set Zn*as<\/p>\n<p style=\"text-align: justify\">as Zn\u2217 = {g1, g2, g3, \u2026, gf(n)}<\/p>\n<p>&nbsp;<\/p>\n<p>Example:<\/p>\n<p style=\"text-align: justify\">The group G = &lt;Z10*, \u00d7&gt; has two primitive roots because f(10) = 4 and f(f(10)) = 2. It can be found that the primitive roots are 3 and 7. The following shows how we can create the whole set Z10* using each primitive root.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-148 aligncenter\" src=\"http:\/\/csp11.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/55\/2018\/07\/Capture-3.jpg\" alt=\"\" width=\"614\" height=\"57\" srcset=\"https:\/\/ebooks.inflibnet.ac.in\/csp11\/wp-content\/uploads\/sites\/55\/2018\/07\/Capture-3.jpg 614w, https:\/\/ebooks.inflibnet.ac.in\/csp11\/wp-content\/uploads\/sites\/55\/2018\/07\/Capture-3-300x28.jpg 300w, https:\/\/ebooks.inflibnet.ac.in\/csp11\/wp-content\/uploads\/sites\/55\/2018\/07\/Capture-3-65x6.jpg 65w, https:\/\/ebooks.inflibnet.ac.in\/csp11\/wp-content\/uploads\/sites\/55\/2018\/07\/Capture-3-225x21.jpg 225w, https:\/\/ebooks.inflibnet.ac.in\/csp11\/wp-content\/uploads\/sites\/55\/2018\/07\/Capture-3-350x32.jpg 350w\" sizes=\"auto, (max-width: 614px) 100vw, 614px\" \/><\/p>\n<p style=\"text-align: justify\">Note that the group G = &lt;Z10*, \u00d7&gt; is always cyclic because p is a prime. The group G =&lt; Z n *, X &gt; is a cyclic group if it has primitive roots. The group G =&lt; Z p*, X &gt; is always cyclic .<\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\"><strong>The idea of Discrete Logarithm<\/strong><\/p>\n<p>&nbsp;<\/p>\n<p>Properties of G = &lt;Zp*, \u00d7&gt; :<\/p>\n<div>\n<p>\u00a0 \u00a0 1.\u00a0 Its elements include all integers from 1 to p \u2212 1.<\/p>\n<p>&nbsp;<\/p>\n<p>2.\u00a0 It always has primitive roots.<\/p>\n<p>&nbsp;<\/p>\n<p>3.\u00a0 It is cyclic. The elements can be created using gx where x is an integer from 1 to f(n) = p \u2212 1.<\/p>\n<\/div>\n<ol start=\"4\">\n<li style=\"text-align: justify\">The primitive roots can be thought as the base of logarithm. If the group has k primitive roots, calculation can be done in k different base. Given x=log, y for any element y is the set, there is another element x that is the log of y in base g. this type of logarithm is called discrete logarithm.<\/li>\n<\/ol>\n<p><strong>\u00a0 \u00a0Solution to Modular Logarithm Using Discrete Logs<\/strong><\/p>\n<p>&nbsp;<\/p>\n<p>Now let us see how to solve problems of type y=ax (mod n) when y is given and we need to find x<\/p>\n<p>&nbsp;<\/p>\n<p><strong>Tabulation of Discrete Logarithms<\/strong><\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\">One way to solve the above mentioned problem is to use a table for each Zp* and different bases.This type of table can be pre calculated and saved. For example, the below table shows the tabulation of the discrete logarithm for Z7*. We know that we have two primitive roots or base in the set.<\/p>\n<p>&nbsp;<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-150\" src=\"http:\/\/csp11.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/55\/2018\/07\/Capture-4.jpg\" alt=\"\" width=\"689\" height=\"97\" srcset=\"https:\/\/ebooks.inflibnet.ac.in\/csp11\/wp-content\/uploads\/sites\/55\/2018\/07\/Capture-4.jpg 689w, https:\/\/ebooks.inflibnet.ac.in\/csp11\/wp-content\/uploads\/sites\/55\/2018\/07\/Capture-4-300x42.jpg 300w, https:\/\/ebooks.inflibnet.ac.in\/csp11\/wp-content\/uploads\/sites\/55\/2018\/07\/Capture-4-65x9.jpg 65w, https:\/\/ebooks.inflibnet.ac.in\/csp11\/wp-content\/uploads\/sites\/55\/2018\/07\/Capture-4-225x32.jpg 225w, https:\/\/ebooks.inflibnet.ac.in\/csp11\/wp-content\/uploads\/sites\/55\/2018\/07\/Capture-4-350x49.jpg 350w\" sizes=\"auto, (max-width: 689px) 100vw, 689px\" \/><\/p>\n<p>&nbsp;<\/p>\n<p><strong>Table 12.1 Discrete logarithm for G = &lt;Z<\/strong><strong>7<\/strong><strong>*, \u00d7&gt;<\/strong><\/p>\n<p>&nbsp;<\/p>\n<p>Example: Find x in each of the following cases:<\/p>\n<ol>\n<li>4 \u2261 3<sup><em>x<\/em><\/sup> (mod 7).<\/li>\n<li>6 \u2261 5<sup><em>x<\/em><\/sup> (mod 7).<\/li>\n<\/ol>\n<div>\n<p>We can easily use the tabulation of the discrete logarithm.<\/p>\n<p>&nbsp;<\/p>\n<p>a. 4 \u2261 3<sup>x<\/sup> mod 7 \u2192 x = L34 mod 7 = 4 mod 7<\/p>\n<p>&nbsp;<\/p>\n<p>b. 6 \u2261 5<sup>x<\/sup> mod 7 \u2192 x = L56 mod 7 = 3 mod 7<\/p>\n<\/div>\n<p><strong>One-way function<\/strong><\/p>\n<p style=\"text-align: justify\">A function f(x) is called a one-way function if f(x) is easy to compute, but, given y, it is computationally infeasible to find x with y=f(x). La(b) is a one-way function if p is large<\/p>\n<p>&nbsp;<\/p>\n<p><strong>Using Properties of Discrete Logarithms<\/strong><\/p>\n<p style=\"text-align: justify\">To see that discrete logarithms behave just like traditional logarithms, several properties of both types of logarithms are given in table 12 .3.Note that the modulus is \u03a6(n) instead of n.<\/p>\n<p>&nbsp;<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"size-full wp-image-151 aligncenter\" src=\"http:\/\/csp11.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/55\/2018\/07\/Capture-5.jpg\" alt=\"\" width=\"571\" height=\"295\" srcset=\"https:\/\/ebooks.inflibnet.ac.in\/csp11\/wp-content\/uploads\/sites\/55\/2018\/07\/Capture-5.jpg 571w, https:\/\/ebooks.inflibnet.ac.in\/csp11\/wp-content\/uploads\/sites\/55\/2018\/07\/Capture-5-300x155.jpg 300w, https:\/\/ebooks.inflibnet.ac.in\/csp11\/wp-content\/uploads\/sites\/55\/2018\/07\/Capture-5-65x34.jpg 65w, https:\/\/ebooks.inflibnet.ac.in\/csp11\/wp-content\/uploads\/sites\/55\/2018\/07\/Capture-5-225x116.jpg 225w, https:\/\/ebooks.inflibnet.ac.in\/csp11\/wp-content\/uploads\/sites\/55\/2018\/07\/Capture-5-350x181.jpg 350w\" sizes=\"auto, (max-width: 571px) 100vw, 571px\" \/><\/p>\n<p style=\"text-align: center\"><strong>Table 12.3 Comparison of traditional and discrete logarithms<\/strong><\/p>\n<p>&nbsp;<\/p>\n<p><strong>Using Algorithms Based on Discrete Logarithms<\/strong><\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\">Tabulation and the properties of discrete logarithms cannot be used to solve y \u2261 ax (mod n) when n is very large. Several algorithms have been devised that use the basic idea of discrete logarithms to solve the problem. Although all of these algorithms are more efficient that the exhaustive-search algorithm that we mentioned at the beginning of this section, none of them have polynomial complexity. Most of these algorithms have the same level of complexity as the factorization problem.<\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\">The discrete logarithm problem has the same complexity as the factorization problem.<\/p>\n<p>&nbsp;<\/p>\n<p><strong>Summary<\/strong><\/p>\n<ul>\n<li>We have learnt about Exponentiation and\u00a0 Logarithm algorithm<\/li>\n<li>\u00a0Observed the need for Order of group and Cyclic Group<\/li>\n<li>\u00a0Learnt about the purpose of Primitive root<\/li>\n<li>\u00a0Understand the Discrete Logarithm problem<\/li>\n<\/ul>\n<table>\n<tbody>\n<tr>\n<td><strong>you can view video on Exponentiation and Logarithm<\/strong><\/td>\n<td><a href=\"https:\/\/youtu.be\/f1chUP70-Gk\" target=\"_blank\" rel=\"noopener\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-120\" src=\"http:\/\/epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/2018\/11\/download.png\" alt=\"\" width=\"36\" height=\"36\" \/><\/a><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"size-full wp-image-153 aligncenter\" src=\"http:\/\/csp11.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/55\/2018\/07\/Capture-6.jpg\" alt=\"\" width=\"467\" height=\"319\" srcset=\"https:\/\/ebooks.inflibnet.ac.in\/csp11\/wp-content\/uploads\/sites\/55\/2018\/07\/Capture-6.jpg 467w, https:\/\/ebooks.inflibnet.ac.in\/csp11\/wp-content\/uploads\/sites\/55\/2018\/07\/Capture-6-300x205.jpg 300w, https:\/\/ebooks.inflibnet.ac.in\/csp11\/wp-content\/uploads\/sites\/55\/2018\/07\/Capture-6-65x44.jpg 65w, https:\/\/ebooks.inflibnet.ac.in\/csp11\/wp-content\/uploads\/sites\/55\/2018\/07\/Capture-6-225x154.jpg 225w, https:\/\/ebooks.inflibnet.ac.in\/csp11\/wp-content\/uploads\/sites\/55\/2018\/07\/Capture-6-350x239.jpg 350w\" sizes=\"auto, (max-width: 467px) 100vw, 467px\" \/><\/p>\n","protected":false},"author":3,"menu_order":11,"template":"","meta":{"_acf_changed":false,"pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-144","chapter","type-chapter","status-publish","hentry"],"part":3,"_links":{"self":[{"href":"https:\/\/ebooks.inflibnet.ac.in\/csp11\/wp-json\/pressbooks\/v2\/chapters\/144","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/ebooks.inflibnet.ac.in\/csp11\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/ebooks.inflibnet.ac.in\/csp11\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/ebooks.inflibnet.ac.in\/csp11\/wp-json\/wp\/v2\/users\/3"}],"version-history":[{"count":5,"href":"https:\/\/ebooks.inflibnet.ac.in\/csp11\/wp-json\/pressbooks\/v2\/chapters\/144\/revisions"}],"predecessor-version":[{"id":548,"href":"https:\/\/ebooks.inflibnet.ac.in\/csp11\/wp-json\/pressbooks\/v2\/chapters\/144\/revisions\/548"}],"part":[{"href":"https:\/\/ebooks.inflibnet.ac.in\/csp11\/wp-json\/pressbooks\/v2\/parts\/3"}],"metadata":[{"href":"https:\/\/ebooks.inflibnet.ac.in\/csp11\/wp-json\/pressbooks\/v2\/chapters\/144\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/ebooks.inflibnet.ac.in\/csp11\/wp-json\/wp\/v2\/media?parent=144"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/ebooks.inflibnet.ac.in\/csp11\/wp-json\/pressbooks\/v2\/chapter-type?post=144"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/ebooks.inflibnet.ac.in\/csp11\/wp-json\/wp\/v2\/contributor?post=144"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/ebooks.inflibnet.ac.in\/csp11\/wp-json\/wp\/v2\/license?post=144"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}